Question

topic 3 characteristics of polynomial functions skills practice continued 2 $m(x) = x^4$ $n(x) = m(x) + 4$ | reference points on $m(x)$ | $
ightarrow$ | corresponding points on $n(x)$ | | :---: | :---: | :---: | | $(0, 0)$ | $
ightarrow$ | | | $(1, 1)$ | $
ightarrow$ | | | $(2, 16)$ | $
ightarrow$ | | (graph of $m(x)$ with points $(-2, 16)$, $(0, 0)$, $(-1, 1)$, $(1, 1)$, $(2, 16)$ on a coordinate grid)

Explanation:

Step1: Analyze the transformation

The function \( n(x) = m(x)+4 \) represents a vertical shift of the function \( m(x) \) by 4 units upward. For a point \((x, y)\) on \( m(x) \), the corresponding point on \( n(x) \) will be \((x, y + 4)\) because we add 4 to the \( y \)-coordinate (since \( n(x)=m(x)+4 \), so when \( m(x)=y \), \( n(x)=y + 4 \)).

Step2: Find the corresponding point for \((0, 0)\)

For the point \((0, 0)\) on \( m(x) \), substitute \( x = 0 \) and \( y = 0 \) into the transformation. The new \( y \)-coordinate is \( 0+4 = 4 \). So the corresponding point on \( n(x) \) is \((0, 4)\).

Step3: Find the corresponding point for \((1, 1)\)

For the point \((1, 1)\) on \( m(x) \), substitute \( x = 1 \) and \( y = 1 \) into the transformation. The new \( y \)-coordinate is \( 1 + 4=5 \). So the corresponding point on \( n(x) \) is \((1, 5)\).

Step4: Find the corresponding point for \((2, 16)\)

For the point \((2, 16)\) on \( m(x) \), substitute \( x = 2 \) and \( y = 16 \) into the transformation. The new \( y \)-coordinate is \( 16+4 = 20 \). So the corresponding point on \( n(x) \) is \((2, 20)\).

Answer:

The corresponding points on \( n(x) \) are:

• For \((0, 0)\): \((0, 4)\)
• For \((1, 1)\): \((1, 5)\)
• For \((2, 16)\): \((2, 20)\)

(If we are filling the table, the table would be:

Reference Points on \( m(x) \)	Corresponding Points on \( n(x) \)
\((1, 1)\)	\((1, 5)\)
\((2, 16)\)	\((2, 20)\)	)